Lecture Notes 1: Matrix Algebra
                             Part B: Determinants and Inverses
                                                 Peter J. Hammond
                                         revised 2018 September 16th
          University of Warwick, EC9A0 Maths for Economists                 Peter J. Hammond                   1 of 87
   Lecture Outline
          Special Matrices
                Square, Symmetric, and Diagonal Matrices
                The Identity Matrix
                The Inverse Matrix
                Partitioned Matrices
          Permutations and Their Signs
                Permutations
                Transpositions
                Signs of Permutations
                Basic Lemma on Signs of Permutations
                Some Implications
          Determinants: Introduction
                Determinants of Order 2
                Determinants of Order 3
                The Determinant Function
                Permutation and Transposition Matrices
                Triangular Matrices
          University of Warwick, EC9A0 Maths for Economists                 Peter J. Hammond                   2 of 87
   Outline
          Special Matrices
                Square, Symmetric, and Diagonal Matrices
                The Identity Matrix
                The Inverse Matrix
                Partitioned Matrices
          Permutations and Their Signs
                Permutations
                Transpositions
                Signs of Permutations
                Basic Lemma on Signs of Permutations
                Some Implications
          Determinants: Introduction
                Determinants of Order 2
                Determinants of Order 3
                The Determinant Function
                Permutation and Transposition Matrices
                Triangular Matrices
          University of Warwick, EC9A0 Maths for Economists                 Peter J. Hammond                   3 of 87
    Square Matrices
              Asquare matrix has an equal number of rows and columns,
              this number being called its dimension.
              The (principal, or main) diagonal
              of a square matrix A = (a )                                          of dimension n
                                            n                        ij   n×n
              is the list (a )                      =(a ,a ,...,a ) of its n diagonal elements.
                                        ii  i=1               11       22                nn
              The other elements a with i 6= j are the off-diagonal elements.
                                                           ij
              Asquare matrix is often expressed in the form
                                                                  a              a          . . .      a 
                                                                         11          12                    1n
                                                                  a              a          . . .      a 
                                                        A= 21                       22                    2n
                                                                   .                .        .            . 
                                                                   .                .          ..         . 
                                                                         .           .                     .
                                                                       a          a          . . .      a
                                                                         n1         n2                     nn
              with some extra dots along the diagonal.
              University of Warwick, EC9A0 Maths for Economists                                       Peter J. Hammond                              4 of 87